By the subadditive property of the SCAD penalty, \(P_{\lambda }(\nu ) \le {} P_{\lambda }(\psi ^{*}) + P_{\lambda }({\widehat{\psi }})\) . Moreover, Lemma 4 of [1]} shows that for the SCAD penalty, \(\lambda |x| \le p_{\lambda }(x) + x^2/(2(a-1))\)  for any \(x\in \mathbb { R}\) . Together, these results give that
\(\frac{\lambda }{2}\Vert \nu \Vert _1 \le {} \frac{1}{2}P_{\lambda }(\nu ) + \frac{1}{4(a-1)}\Vert \nu \Vert _{2}^2 \le {} \frac{P_{\lambda }(\psi ^{*})}{2} +\frac{P_{\lambda }({\widehat{\psi }})}{2} + \frac{1}{4(a-1)}\Vert \nu \Vert _{2}^2.\) 
